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# Full text of "Treatise On Analysis Vol-Ii"

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```368       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

constant y > 0 such that

(15.8.14.1)                        V(x\y) = (Ufa) \Ufry)

(for the scalar product defined in (15.7.4)).

If A is finite-dimensional, then the image of A under Ul is the algebra
Endc(I) of all endomorphisms of the vector space I, and the relation (15.8.14.1)
remains valid, for the scalar product defined in (15.7.4.1) (from the scalar
product on I -which is the restriction of that on A).

We can assume that A is the Hilbert sum of a (finite or infinite) sequence of
minimal left ideals !„ = Aen, where I = Ix (15.8.11.1) and all the In are isomor-
phic (15.8.13). If x T£ 0 and Ufa) = 0, that is, if xl = {0}, we should have
(Ax)l = {0}, and since the ideal Ax is nonzero, it contains a minimal left ideal
I7 (15.8.8), which must be isomorphic to I (15.8.13); hence I'l = {0}, contrary
to (15.8.12(iii)). Hence the representation £/r is faithful. Put Pn = t/i(en),
which is the orthogonal projection of I = Ael onto the one-dimensional subspace
en A*i (15.8.12), because (xe1 - enxel \ enyej = (enxei - el xel \ye^ = 0.
Since emen = 0 if m ^ n, we have PmPn=0 and therefore the subspaces
enAei are orthogonal in pairs. Moreover, I is the Hilbert sum of the sub-
spaces enAel9 because, if xe1 is orthogonal to all these subspaces, we have
Pn(xe1) = 0 for all n, so that enxe± = 0 for all «, and therefore xe1 belongs
to the right annihilator of A, which is zero (15.7.5.7). This shows that the
sequence (ln) is finite if and only if I (and therefore each In) is finite-dimQn-
sional over C, or equivalently if and only if A is finite-dimensional.

Let (an) be an orthonormal basis of I such that an e en Ael for each n.
Then anafeenAen, hence ana* = !„<?„ for some AB e C* (15.8.12). Likewise
We have ^ = An, because on the one hand

and on the other hand

«!• anan a* = % *n el<*n   = *>'n ^n * * = ^ An en ,

since an et = an. Also

1 = fa, I flj = (a,, I en an) = (ana* \ en) = Xn(en \ en)
and

1 = (fl. IO = (^ I a, cx) = (a*aM | ej = An(^ | ej;
hence

(15.8.14.2)                       •feU) = fc|e,)ce also contains
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